Towards a Theoretical Understanding of Why Local Search Works for Clustering with Fair-Center Representation

The representative k-median problem generalizes the classical clustering formulations in that it partitions the data points into several disjoint demographic groups and poses a lower-bound constraint on the number of opened facilities from each group, such that all the groups are fairly represented by the opened facilities. Due to its simplicity, the local-search heuristic that optimizes an initial solution by iteratively swapping at most a constant number of closed facilities for the same number of opened ones (denoted by the O(1)-swap heuristic) has been frequently used in the representative k-median problem. Unfortunately, despite its good performance exhibited in experiments, whether the O(1)-swap heuristic has provable approximation guarantees for the case where the number of groups is more than 2 remains an open question for a long time. As an answer to this question, we show that the O(1)-swap heuristic (1) is guaranteed to yield a constant-factor approximation solution if the number of groups is a constant, and (2) has an unbounded approximation ratio otherwise. Our main technical contribution is a new approach for theoretically analyzing local-search heuristics, which derives the approximation ratio of the O(1)-swap heuristic via linearly combining the increased clustering costs induced by a set of hierarchically organized swaps.